Operator ideal norms on

Authors:
L. Rodríguez-Piazza and M. C. Romero-Moreno

Journal:
Trans. Amer. Math. Soc. **352** (2000), 379-395

MSC (1991):
Primary 47D50, 46E30

DOI:
https://doi.org/10.1090/S0002-9947-99-02196-0

Published electronically:
July 20, 1999

MathSciNet review:
1473454

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a real number such that and its conjugate exponent . We prove that for an operator defined on with values in a Banach space, the image of the unit ball determines whether belongs to any operator ideal and its operator ideal norm. We also show that this result fails to be true in the remaining cases of . Finally we prove that when the result holds in finite dimension, the map which associates to the image of the unit ball the operator ideal norm is continuous with respect to the Hausdorff metric.

**[AD]**Anantharaman, R. and Diestel, J.,*Sequences in the range of a vector measure*, Comment. Math. Prace Mat.**30**(1991), 221-235. MR**92g:46049****[B]**Bolker, E. D.,*A class of convex bodies*, Trans. Amer. Math. Soc.**145**(1969), 323-345. MR**41:921****[DJT]**Diestel, J., Jarchow, H. and Tonge, A.,*Absolutely summing operators*, Cambridge studies in advance mathematics 43, Cambridge, 1995. MR**96i:46001****[DU]**Diestel, J. and Uhl Jr., J.J.,*Vector Measures*, Amer. Math. Soc. Surveys 15, Providence, R. I., 1977. MR**56:12216****[G]**Grothendieck, A.,*Produits tensoriels topologiques et espaces nucléaires*, Mem. Amer. Math. Soc. 16 (1955). MR**17:763c****[K]**Kanter, M.,*The -norm of sums of translates of a function*, Trans. Amer. Math. Soc.**179**(1973), 35-47. MR**50:14062****[L]**Linde, D.R.,*Moments of measures in Banach spaces*, Math. Ann.**258**(1982), 277-287. MR**83f:60010****[Lu]**Lusky, W.,*Some consequences on Rudin's paper "-isometries and equimeasurability"*, Indiana University Math. J.**27**(1978), 859-866. MR**81h:46028****[N]**Neyman, A.,*Representation of -norms and isometric embedding in -spaces*, Israel J. Math.**48**(1984), 129-138. MR**86g:46033****[Pe]**Pe{\l}czynski, A.,*-integral operators commuting with group representations and examples of quasi--integral operators which are not -integral*, Studia Math.**33**(1969), 63-70. MR**39:6125****[P]**Pietsch, A.,*Operator ideals*, North-Holland, Amsterdam, 1980. MR**81j:47001****[R1]**Rodríguez-Piazza, L.,*The range of a vector measure determines its total variation*, Proc. Amer. Math. Soc.**111**(1991), 205-214. MR**91e:46053****[R2]**Rodríguez-Piazza, L.,*Derivability, variation and range of a vector measure*, Studia Math.**112**(1995), 165-187. MR**96c:28014****[RR]**Rodríguez-Piazza, L. and Romero-Moreno, M.C.,*Conical measures and properties of a vector measure determined by its range*, Studia Math. 125 (1997), 255-270. MR**98i:46042****[Ru1]**Rudin, W.,*Projections on invariant subspaces*, Proc. Amer. Math. Soc.**13**(1962), 215-228. MR**25:1460****[Ru2]**Rudin, W.,*-isometries and equimeasurability*, Indiana University Math. J.**25**(1976), 429-432. MR**53:14105**

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Additional Information

**L. Rodríguez-Piazza**

Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, Sevilla 41080, Spain

Email:
piazza@cica.es

**M. C. Romero-Moreno**

Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, Sevilla 41080, Spain

Email:
mcromero@cica.es

DOI:
https://doi.org/10.1090/S0002-9947-99-02196-0

Keywords:
Operator ideals,
ideal norm,
$L^p$ spaces,
Hausdorff metric,
$p$-integral operators

Received by editor(s):
May 30, 1997

Published electronically:
July 20, 1999

Additional Notes:
Research supported in part by DGICYT grant #PB93–0926

Article copyright:
© Copyright 1999
American Mathematical Society